Problem 1. (Optimal reserve prices) In class, we had a brief discussion of optimal auction design. In this problem we will explore this direction. In this problem, we assume two bidders compete to buy a single indivisble good, in a second price auction with a reserve price R set by the auctioneer. The auction works as follows. Each bidder has a private valuation v1, 02 that is uniformly distributed on [0, 1]; each bidder only observes their own valuation. After observing their valuation, each bidder chooses a bid; let 1;,- denote the bid of bidder i. If both bids are higher than the reserve, i.e., b1, b2 2 R, then the higher bidder wins and pays the second highest bid. (Assume all ties are broken in favor of bidder 1 winning the auction.) If one bid is higher than R (i.e., max{b1, b2} 2 R), but the other bid is lower than R (i.e., min{b1, b2}